Answer:
, assuming that the rollercoaster is not attached to the track.
Step-by-step explanation:
Let
denote the radius of the loop. The height of the top of the loop would be
.
Let
denote the acceleration of the rollercoaster. At any position in the loop, if the speed of the rollercoaster is
, the (centripetal) acceleration of the rollercoaster would be:
.
Let
denote the mass of the rollercoaster. The net force on the rollercoaster would then be:
.
The rollercoaster would stay on the track (and goes around the loop without falling off) only if the normal force
between the track and the rollercoaster is non-negative. In other words: it is necessary that
for the rollercoaster to stay on the track.
At the top of the loop,
and the weight of the rollercoaster
are in the same direction as the centripetal acceleration (downwards towards the center of the loop.) Hence:
.
Let
denote the speed of the rollercoaster at the top of the loop.
.
If
at the top of the loop, then:
.
.
At the same time, by the conservation of energy, the sum of the kinetic energy
and gravitational potential energy
of the rollercoaster should stay the same during the entire ride. Assuming that
is
at the bottom of the loop:
.
Let
denote the speed of the rollercoaster at the bottom of the loop.
.
The speed of the rollercoaster at the top of the loop
is at least
. Therefore:
.
Since the height of the loop is
, the
of the rollercoaster at the top of the loop would be:
.
Thus:
.
In other words:
.
Rearrange and simplify to obtain a bound on
:
.
.
Given that
and
:
.
Hence, the radius of this loop is at most
, such that the height of the top of this loop is at most
.